1. Different Monitorings are done by Availability Mgmnt, Capacity mngmnt, SLM etc. Are these monitorings done via Service Desk or directly by people involved in these processes?

AM certainly utilizes SD IM data to analyze availability, as would SLM though more after the fact via reports, SOAs, analysis and such. CpM would generally be doing their thing utilizing many different inputs as well as possibly the IM data. AM and SLM should work very closely around achievement of agreed upon service levels as would CpM and AM as capacity issues can often impact availability. SD may have a role in monitoring, responding to automated alerts which auto-generate Incidents for example, but your language is pretty broad here. Normally SD would not be "doing" the AM, SLM or CpM monitoring activities though. A suggestion for when you reach a confusion point on questions like this: remember the basic difference between Service Support activity and Service Delivery activity... SLM, AM and CpM are Service Delivery processes. SD is not.

3. Who finds the structural resolution of the Error. Problem control after identfying the known error or Error control after receiving the informaiton about the known error from Problem control.

If I understand you correctly, you are doing a common thing here by confusing activities in PM with people (who). IME it is most often the same person (someone in a PM group) doing all these things or at least overseeing/owning them with others possibly doing the actual discovery of or implementation of a permanent correction. If you are in fact asking within the context of the PM process, the answer is it could be either. A KE is an Error with a known cause or one with a workaround identified. If you mean discovering the actual underlying cause, it could be known by Problem Control along with a work around or it could be determined by Error Control. Error Control wants to resolve the KE via CM. Problem Control wants to progress the Problem to a KE. Remember, these are roles and activities, not necessarily people or different people or departments. This is a somewhat fuzzy area though so I appreciate your confusion. If this were on a test, I would go with Error Control and then complain about the question.

4. Who is in charge of DSL? Configuration Management or Release Management?

Release Management

5. In the Configuration Management chapter of blue book, only DSL is mentioned. Does configuration Management have to do anything with DHS also or not?

CfM would maintain a record in the CMDB for status, attributes, location, etc. of CIs in the DHS. RM manages the what and where of the DHS (and DSL).